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Differentiation Applications 2: Linear Approximation
159
Pre-Assignment for Linear Approximation
Instructions: Print out the activity. Complete questions 1 through 3 of the activity.
Once you have completed this pre-assignment, go to the Canvas page for your class and
complete the pre-assignment quiz that you will find there. You have as many tries as you
like for each question. This must be completed BEFORE your peer led session on Friday.
Bring your printed activity with your completed pre-assignment to your peer led session in
order to be eligible to take the quiz that will occur at the beginning of your peer led session.
Your quiz will be based on last week’s activity (unless you are in Dr. Burgos’ class, in which
case your peer leaders will tell you what to do when you get to class.)
If you need help completing the pre-assignment, feel free to drop in at SMART lab (at the
library tutoring services).
SMART Lab Hours are:
M – Th:
F:
Sa:
Su:
9am – 9pm
9am – 4pm
closed
1 – 5pm
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Differentiation Applications 2: Linear Approximation
DA 2: Linear Approximation
Model 1: Sleepy Road Trip
You are on a road trip with your family and have been traveling for nearly two hours on small roads
when you get on the highway. Soon after entering the highway, you fall asleep. The trip odometer
was reset to zero at the start of the trip, and right before you fell asleep you notice it read 60 km, and
the speedometer read 60 km/hour (1 km/min).
Construct Your Understanding Questions (to do in class)
1. With only the information in Model 1, make your best estimate of what the trip odometer will
read when you wake if you sleep for 60 minutes. Explain your reasoning.
2.
f (t ) is the distance of the car from the starting point after t minutes. If you fell asleep at
t = 120 minutes, and ∆t is the length of time you were asleep, complete the following table.
Symbol
Units
∆t
minutes
Value
Description
length of time you were asleep
f (120)
f ′(120)
km/min
f (120 + ∆t )
Estimate this based on your
answer to Question 1
(actual value is unknown)
3. Use words to fill in the blanks in the following sentence about the Sleepy Road Trip:
“The trip odometer reading at the moment I woke is approximately equal to the trip
odometer reading at the moment _________________________ plus the speed of the car
at the moment __________________________ × ___________________________.”
4. Use symbols from the table to turn the sentence in the previous question into an
expression for…
f (120 + ∆t ) ≈
5. (Check your work) Show how the expression you generated in the previous question was (or
can be) used to answer Question 1. If your expression does not work in Question 1, go back
and reexamine your answers to the previous two questions.
Differentiation Applications 2: Linear Approximation
6. The graph at right shows what happened
during the Sleepy Road Trip in Model 1.
b. (Check your work) Does the point
labeled “60 min nap estimate” match
your answer to Question 1? If not, go
back and reconsider Question 1.
y
180
120
y = f (t )
60 min nap
estimate
km
a. Mark and label the point on the graph
that corresponds to the time at which
you fell asleep.
161
60
t
c. Mark the point on the graph
corresponding to the actual odometer
reading the moment you woke from
your 60 minute nap.
60
120
180
240
3
minutes
d. The distance between the actual and estimated odometer readings is the error in this
estimate. Mark this vertical gap on the graph and label it E (180) since it is the Error in
your estimate at t = 180 . Based on the graph, the value of E (180) = __________km.
e. Why was the estimate off? What incorrect assumption was built into the estimate you
made in Question 1?
7. On the graph above, draw a line from the point you labeled “fell asleep” to the point labeled
“60 min nap estimate.”
a. Describe the difference between the imaginary trip represented by this line versus the
actual trip represented by the function f during the interval from t = 120 to 180 .
b. This line is the tangent line to the graph of y = f (t ) at t = ________ minutes.
c. Use the graph to find the slope of this tangent line. Show your work.
d. Which of the following expressions best describes this slope? [circle one]
f ′(120)
120 − f (120)
f ′(180)
f (180) − f (120)
8. The estimate you made in Question 1 is called a linear approximation. Construct an
explanation for why this type of approximation is called a linear approximation.
162
Differentiation Applications 2: Linear Approximation
9. Use a linear approximation based on the position and speed of the car at t = 120 to estimate
the position of the car at t = 140 . (Write down any assumptions and show your work.)
a. On the graph in Question 6, mark a point that corresponds to your linear
approximation of the position of the car at t = 140 .
(Check your work) This is not the point on the curve at (140,85) .
b. What is the error in this approximation? E (140) = ________km
10. Fill in the blanks: By analogy to Model 1, the linear approximation in the previous question
corresponds to a nap starting at t = ________ minutes, and lasting _________ minutes.
11. In this activity you used the position and speed of the car in Model 1 at t = 120 to estimate
the car’s position at t = 140 and at t = 180 . Even without the graph of y = f (t ) shown in
Question 6, one would guess that the estimate at t = 140 is more accurate than the estimate
at t = 180 . Explain.
12. Give an example of a value of t for which a linear approximation using the tangent line to f
at t = 120 on the previous page would give an error smaller than either of the estimates we
have made so far (that is, smaller than E (180) or E (140) ).
13. In general, the smaller or larger [circle one] the value of ∆t in f (t + ∆t ) ≈ f (t ) + f ′(t ) ⋅ ∆t ,
the smaller the error in corresponding linear approximation. (Check your work) Is this
consistent with your answers to the previous two questions? Explain.
Differentiation Applications 2: Linear Approximation
163
Activity Report
Linear Approximation
We verify that we all understand and agree with the solutions to these questions.
Group Number: _______
Manager: _____________________________________________
Recorder: _____________________________________________
Spokesperson: _________________________________________
Strategy Analyst: _______________________________________
Critical Thinking Question: to be agreed upon by the group, and written below by the recorder.
Use your work in Model 1 and a linear approximation based on the position and speed of the car at t = 120
minutes to estimate the position of the car at t = 130 minutes. Explain your work. Be sure to write complete
sentences, and include your printed first and last names, and UIDs.
For instructor’s use only
All questions on activity filled out
Names and U-Numbers PRINTED on activity report
Critical thinking correct, fully justified, and written in complete sentences
Satisfactory/Unsatisfactory
…
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